This thesis disentangles Gauss-Newton and variational approximations in Bayesian deep learning.
problem Understanding the interplay between the Gauss-Newton method and variational approximations in Bayesian deep learning.
method Analysis of the Gauss-Newton method and Laplace/Gaussian variational approximations for neural networks.
result The combination of the Gauss-Newton method with approximate inference can be cast as inference in a linear or Gaussian process model.
The Gauss-Newton method is analyzed for neural networks using Riemannian optimization techniques.
problem Training neural networks with smooth activations and convergence rates.
method Riemannian optimization perspective, analyzing the Gauss-Newton method in both underparameterized and overparameterized regimes.
result Geometric convergence rates independent of conditioning and eigenvalues, demonstrating accelerated convergence.
A new method for machine learning updates reduces complexity and improves robustness.
problem Stochastic gradient updates are inefficient and sensitive to feature scaling.
method Incremental Gauss-Newton Descent (IGND) reduces the need for matrix operations and improves robustness.
result IGND improves robustness to sensitivity scaling and can be competitive with common stochastic optimizers.
EGN optimizes deep neural networks with exact Gauss-Newton for large-scale problems.
problem Training deep neural networks efficiently and accurately.
method Stochastic second-order optimization using low-rank linear algebra and matrix factorization.
result Converges to stationary points of the objective function under mild assumptions.
A new optimization method improves deep learning accuracy without hyper-parameter tuning.
problem Computational demands and convergence behavior in deep learning training.
method Stochastic quasi-Gauss-Newton (SQGN) optimization method combining stochastic quasi-Newton, Gauss-Newton, and variance reduction.
result SQGN provides excellent accuracy without hyper-parameter experimentation, improving convergence and computational performance.
New method improves generalization in deep learning models.
problem Improving generalization in overparameterized deep neural networks.
method Stochastic Gauss-Newton method with Levenberg-Marquardt damping and mini-batch sampling.
result Established finite-time convergence and non-asymptotic generalization bounds.
We present practical Levenberg-Marquardt variants of Gauss-Newton and natural gradient methods for solving non-convex optimization problems that arise in training deep neural networks involving enormous numbers of variables and huge data sets. Our methods use subsampled Gauss-Newton or Fisher information matrices and e…
The study characterizes the conditioning of the Gauss-Newton matrix in neural networks.
problem Understanding the conditioning of the Gauss-Newton matrix in neural networks.
method Theoretical analysis of the GN matrix in deep linear and ReLU networks, extending to residual connections and convolutional layers.
result Established tight bounds on the condition number of the GN matrix in neural networks.
A new method for 1-bit matrix completion that is faster and more accurate.
problem Estimating a low-rank matrix from binary observations.
method Majorization-Minimization Gauss-Newton (MMGN) method.
result MMGN outperforms existing methods in accuracy and speed.
Approximate Newton methods are a standard optimization tool which aim to maintain the benefits of Newton's method, such as a fast rate of convergence, whilst alleviating its drawbacks, such as computationally expensive calculation or estimation of the inverse Hessian. In this work we investigate approximate Newton meth…
We develop two new stochastic Gauss-Newton algorithms for solving a class of non-convex stochastic compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions, and use both classical stochastic and SARAH estimators for approxima…
Stochastic SGN method converges faster than SGD for DNNs.
problem Training deep neural networks efficiently.
method Stochastic generalized Gauss-Newton method using conjugate gradient and automatic differentiation.
result SGN requires fewer iterations and is more robust to hyperparameters.
Bayesian deep learning avoids underfitting by projecting onto null space of generalized Gauss-Newton matrix.
problem Bayesian deep learning often underfits, leading to less accurate predictions than point estimates.
method Proposes a matrix-free algorithm to project onto the null space of the generalized Gauss-Newton matrix, ensuring Bayesian predictions do not underfit.
result The method scales to large models, including vision transformers with 28 million parameters, and avoids underfitting.
ViViT efficiently computes curvature for deep networks without approximations.
problem Efficiently computing curvature for deep networks without approximations.
method Leverages the GGN's low-rank structure without further approximations.
result ViViT allows for efficient computation of eigenvalues, eigenvectors, and directional derivatives.
Ginger efficiently approximates curvature with linear complexity for neural networks.
problem Quadratic memory and cubic time complexity for computing curvature matrices in deep learning.
method Ginger uses eigendecomposition to maintain the inverse of the generalized Gauss-Newton matrix, achieving linear memory and time complexity.
result Ginger provides an effective and efficient curvature approximation for non-convex objectives.
Paper develops RGN method for estimating low-rank tensors from noisy measurements.
problem Estimating low-rank tensors from noisy linear measurements.
method Riemannian Gauss-Newton (RGN) method for efficient low-rank tensor estimation.
result First local quadratic convergence guarantee of RGN for low-rank tensor estimation in noisy settings.
In this paper, we develop a variant of the well-known Gauss-Newton (GN) method to solve a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to the standard GN method, our algorithm allows one to handle general smooth convex objective function. We show, under mild conditions, that …
We present an efficient block-diagonal ap- proximation to the Gauss-Newton matrix for feedforward neural networks. Our result- ing algorithm is competitive against state- of-the-art first order optimisation methods, with sometimes significant improvement in optimisation performance. Unlike first-order methods, for whic…
A new method for robust matrix completion overcomes limitations of existing approaches.
problem Robust matrix completion from corrupted entries, especially under overparameterization and ill-conditioning.
method Factorization-based iterative algorithm combining Gauss-Newton linearization and outlier removal.
result Theoretical guarantees of exact recovery for suitable assumptions.
New framework allows selective removal of stale data in option calibration.
problem Inability to remove old data from calibrated option pricing models without full retraining.
method Introduces operator-theoretic Gauss-Newton framework for selective forgetting.
result Provides stability guarantees and perturbation bounds for selective data removal.
This work approximates full conformal prediction for neural networks without sample splitting.
problem Uncertainty quantification for neural network regression models.
method Approximating full conformal prediction using Gauss-Newton influence for post-hoc uncertainty estimation.
result Locally-adaptive and often tighter prediction intervals compared to split-CP.
New method estimates marginal likelihood for deep learning models using training data alone.
problem Estimation difficulties in marginal likelihood for model selection in deep learning.
method Scalable marginal likelihood estimation based on Laplace's method and Gauss-Newton approximations.
result Estimate outperforms cross-validation and manual tuning on various datasets.
ADCMs adaptively discretize CMs for efficient training.
problem Manual discretization schemes cause repeated adjustments for different noise schedules and datasets.
method Unified framework with optimization problem, local and global consistency constraints, and Gauss-Newton method.
result Significantly improve training efficiency and generative performance of CMs.
First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the prohibitive computational cost in calculating the second-order information. In this pa…
Theoretical framework for target propagation shows differences from backpropagation.
problem Understanding and improving target propagation for neural networks.
method Mathematical optimization analysis and novel reconstruction loss.
result A novel reconstruction loss improves feedback weight training and introduces architectural flexibility.
Enhanced VMC methods improve neural wavefunction training.
problem Efficiently training neural wavefunctions in VMC to converge to energy minimum.
method Rayleigh-Gauss-Newton (RGN) optimization and parallel tempering sampling.
result RGN method achieves superlinear convergence with reduced computational cost.
New algorithms improve RPCA for large matrices with upper rank bounds.
problem Efficiently decompose large matrices into low-rank and sparse parts.
method Combine regularization and matrix multiplication approaches with upper rank bounds.
result Proposed algorithms are faster and more robust than existing methods.
We propose a modular extension of backpropagation for the computation of block-diagonal approximations to various curvature matrices of the training objective (in particular, the Hessian, generalized Gauss-Newton, and positive-curvature Hessian). The approach reduces the otherwise tedious manual derivation of these mat…
In this paper, we provide an overview of first-order and second-order variants of the gradient descent method that are commonly used in machine learning. We propose a general framework in which 6 of these variants can be interpreted as different instances of the same approach. They are the vanilla gradient descent, the…
GNvPro efficiently trains neural networks with variable projection, improving accuracy and generalization.
problem Optimizing weights of neural networks for accurate approximation of input-target data.
method Variable projection (VarPro) extended to non-quadratic objectives, using Gauss-Newton method (GNvPro).
result GNvPro solves optimization problems more efficiently and finds better generalizing solutions.
Regularizes 3D inverse scattering with tangent-point energy for better solutions.
problem Ill-conditioned inverse obstacle scattering problems in 3D.
method Tikhonov regularization using tangent-point energy to penalize surface roughness and ensure well-posedness.
result Regularized solutions converge to true solution as noise level decreases.
Kalman filtering and smoothing algorithms are used in many areas, including tracking and navigation, medical applications, and financial trend filtering. One of the basic assumptions required to apply the Kalman smoothing framework is that error covariance matrices are known and given. In this paper, we study a general…
Second-order methods for neural network optimization have several advantages over methods based on first-order gradient descent, including better scaling to large mini-batch sizes and fewer updates needed for convergence. But they are rarely applied to deep learning in practice because of high computational cost and th…
The Hessian of neural networks can be decomposed into a sum of two matrices: (i) the positive semidefinite generalized Gauss-Newton matrix G, and (ii) the matrix H containing negative eigenvalues. We observe that for wider networks, minimizing the loss with the gradient descent optimization maneuvers through surfaces o…
Deep network improves electrical tomography across multiple frequencies.
problem Nonlinear multi-frequency electrical impedance tomography (mfEIT) for tissue conductivity estimation.
method Integrates graph neural networks (GNNs) into the iterative Proximal Regularized Gauss Newton (PRGN) framework to reconstruct tissue concentrations accurately.
result Accurate reconstruction of overlapping tissue fraction concentrations across multiple frequencies.
New tensor recovery method uses Riemannian optimization on Segre manifold.
problem Recovering low-rank tensors from noisy measurements.
method Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN) algorithms over the Segre manifold.
result Proven convergence rates for RGD and RGN under mild noise assumptions.
Deep learning involves a difficult non-convex optimization problem, which is often solved by stochastic gradient (SG) methods. While SG is usually effective, it may not be robust in some situations. Recently, Newton methods have been investigated as an alternative optimization technique, but nearly all existing studies…
Unified approach to Bayesian inference with guarantees on covariance matrices.
problem Approximate Bayesian inference with PSD guarantees.
method Bayes-Newton methods extending Newton's method for optimisation.
result Novel algorithms with PSD covariance matrices.
New method improves calibration of neural networks by targeting robust margins and local smoothness.
problem Poor calibration of neural networks, leading to unreliable confidence estimates.
method Intervene on training procedure by targeting robust margins and local smoothness.
result Improved out-of-sample calibration without sacrificing accuracy.
New method accelerates neural network training by focusing on flat directions.
problem Improving neural network training speed and stability.
method Bulk-SGD, interpolated gradient methods.
result Updates along the Dominant subspace can accelerate convergence but compromise stability.
The paper studies efficient Hessian fitting methods for stochastic optimization.
problem Efficient Hessian fitting for stochastic optimization.
method Preconditioned Stochastic Gradient Descent (PSGD) method and Lie groups.
result Hessian fitting problem is strongly convex in certain Lie groups.
Unified framework for sampling from complex densities using PDEs and neural networks.
problem Sampling from complicated probability densities.
method Dynamical measure transport via PDEs and physics-informed neural networks (PINNs).
result Significantly better mode coverage and high accuracy in sampling.
This paper improves Bayesian neural nets by using local linearization.
problem Underfitting in Bayesian neural networks.
method Local linearization of Bayesian neural networks to create a generalized linear model (GLM) for predictions.
result The GLM predictive resolves common underfitting problems of the Laplace approximation.
Shampoo optimizes preconditioners for faster convergence in machine learning.
problem Improving convergence speed in machine learning optimization.
method Explicit connection between Shampoo's Kronecker product approximation and optimal matrix approximations.
result The square of Shampoo's approximation is equivalent to a single power iteration step for optimal Kronecker product approximation.
New algorithm reduces rank constrained optimization problems.
problem Rank constrained optimization problems in machine learning and statistics.
method Recursive Importance Sketching (RISRO) algorithm.
result RISRO offers clear advantages over existing algorithms and converges efficiently.
The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. PDE-constrained LDDMM methods are formulated as constrained variational problems, wher…
High-dimensional models trained on smooth manifolds achieve optimal rates in Wasserstein metrics.
problem Training score-based generative models on complex, low-dimensional manifolds.
method Proves optimal rates for SGMs on smooth manifolds, separating into noise regimes and using ReLU nearest-projection coordinates.
result Optimal intrinsic Wasserstein rates are achieved, with polynomial ambient dependence for families with controlled geometry and density.
Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of 2…